I took this solution from the following link : http://forums.linuxmint.com/viewtopic.php?f=53&t=145120
Credits go to the original Author.
This is for BCM4312 chipset only (as stated in the original post). To be very specific, this is for BCM4312 LPPHY chipset.
I am re writing the same because it was very difficult for me to search and obtain this answer when I had the same problem, and this is the only solution that helped. Hope, posting it in a detailed manner helps!
A lot of suggestions were to open the driver Manager (Linux Mint 15) and select the Broadcom device for usage. (But this only resulted in Kernel panic as soon as I connected to Wi-Fi and started using it. May be this is because my chipset was BCM4312 in the Dell Inspiron. )
DO NOT DO THIS IF YOUR CHIPSET IS BCM4312! (You can identify what chipset your wireless is, using the hardware information software which came with Linux Mint KDE.)
I did a fresh upgrade and did the following as suggested in the link above (You may not need to do the fresh upgrade for this to work. I was formatting my hard disk so I didn’t mind installing it again. ):
Note: You need to be connected to the internet through Ethernet or USB tethering for the commands to download packages and install them.
sudo apt-get remove --purge bcmwl-kernel-source
sudo apt-get install firmware-b43-lpphy-installer
Restart the computer and Voila! The wireless is working. Thanks to the author!
I almost felt like moving to another linux distro because of wireless problems in Linux Mint. Thanks to this trick, I can stay.